A Note on the Arrow-Lind Theorem

Kenneth Arrow and Robert Lind have recently proved a theorem on risky public projects, stating that under certain conditions the social cost of the risk tends to zero as the population tends to infinity, so that projects can be evaluated on the basis of expected net benefit alone. The present note gives an alternative formulation and a short new proof of the theorem, and uses these to examine the role of certain assumptions concerning the operation of the public sector which in the original were left implicit or received inadequate attention. Some general critical comments on the applicability of the theorem are also offered. The conditions stated by Arrow and Lind as sufficient for the validity of their result include the following: (i) the government initially appropriates all benefits and pays all costs, distributing the net returns subsequently "through changes in the level of taxes" (p. 371); (ii) the net returns are statistically independent of each person's disposable income in the absence of the project; and (iii) each person's share of the net returns tends to zero as the number n of persons tends to infinity. The result is proved formally only for the case where "all taxpayers [are] identical in that they [have] the same utility function, their incomes [are] represented by identically distributed variables, and they [are] subject to the same tax rates"; but the authors state that " . . .the basic theorem still holds, provided that as n becomes larger the share of the public investment borne by any individual becomes arbitrarily smaller" (p. 373). This theorem, if generally applicable, would have important practical consequences. It would tend to support an extension of public sector investment by justifying the use of a riskless